The best I can come up with is that you are solving a homogenous equation for the non-forced system response. The solution is a linear combination of exponentials.
you found
\[ e^{jw_0} = 2 \text{ and } e^{jw_1} =3 \]
the solution would be
\[y= c_0 e^{jw_0n} +c_1 e^{jw_1n} \]
which becomes
\[y= c_0 2^{n} +c_1 3^{n} \]